Nuclear Physics Bl14 (1976) 147-156 ? North-Holland Publishing Company A MODEL FOR CHARGED SECOND CLASS CURRENTS * M.-S. CHEN, F.S. HENYEY ** and G.L. KANE Randall Laboratory of Ph3;sics, University of Michigan, Ann Arbor M1 48109, USA Received 6 July 1976 Choosing a model for the second class axial current composed of an s-wave vector current pseudoscalar meson pair to fix commutation relations, and using vector domi- nance (B-meson) techniques to calculate, we use recent experimental results to estimate the coupling strength of the second class current. Remarkably, within experimental errors we find that this coupling strength is the same as for the first class current, with the most natural normalization of the second class current. We comment on B-production and production by neutrinos. 1. Introduction A recent experiment \[1 \] has reported a non-zero signal for second class currents \[2\] in the/3-decyas of several nuclear species. Second class currents have not yet, however, been observed in any particle physics experiment. In this paper we investi- gate some phenomenological consequences for ?article physics of a particular model for the second class current. Remarkably, withir the framework of the model, we are led to a second class current which couples to, the weak interactions with the same strength as the ordinary current, when the normalization is chosen in a particu- larly simple way. Second class currents do not naturally appear in the quark model. If they are large it could considerably complicate weak interaction theory; in particular, the weak and electromagnetic interactions could no longer generate only an SU(2) X U(1) as they do in the conventional theory. The second class current \[3-5\] *** by definition has the value -1 for the quan- t~lm number ((-1)IGp)((-1)J+I), while the first class current has the value +I for this quantum number. The odd value for this combination is usually described as hav- ing the opposite G to the usual current. An ffpair, where fis a spin-? fermion, has an *Research supported in part by the U.S. Energy Research and Development Agency. **Address after July 1 : Department of Physics, University of California, San Diego, La Jolla CA 92037. ***Refs. \[3-5\] give good recent reviews of different approaches to second class currents and of varous consequences of second class currents. 147 148 M.-S. Chen et al. / Second class currents even (odd) value of (-)/GP if the pair is in a spin triplet (singlet). Therefore orbital angular momentum is required to make a second class current from such a fermion pair. It turns out that the orbital angular momentum carried by the lower components of the Dirac spinor does not suffice to change the value of (_)I+J+l GP in the vector or axial currents; so it is necessary to have a derivative coupling to make a second class current f~om a spin -1 fermion pair. This need for a derivative coupling is another common way of stating the definition of a second class current. As an alternative to the derivative coupling, a second class current can be con- structed by using more than a pair of fermions, for example two pairs. A model of this latter type was proposed by Lipkin \[6\], who suggested that the second class cur- rent should be made of a first class current and a pion in a relative s-wave, with a sym- metric isospin coupling: 12 = 1 - I 1 . In the SU(3) extension the pseudoscalar octet is coupled in the appropriate way. This suggestion was extended for neutral currents by Adler, Dashen, Healy, Kar- liner, Lieberman, Ng, and Tsao \[3\], who calculate the commutation relations involv- ing the second class currents in the e-model, and replace the o-field in the commuta- tor by its vacuum expectation value. The model we adopt in this paper is the SU(3) generalization of thai of Adler et al. Some comparison with other models will be made at the end of the calculation. Moreover, we shall assume that the axial charged nonstrange second class current is vector meson (B) dominated. We do believe, however, that our most important re- sults are independent of the B-dominance assumption. There is a major difference between the way our charged current is included in the weak interactions and the way Adler et al. 's neutral current is. They suggest an additional neutrino in order to conserve CP. The charged counterparts of their cur- rent could be present (if the new neutrino had enough mass to prohibit second class mu decay), but would not be the current we are discussing. The reason is that the phenomena we are concerned with result from the interference between first and second class currents, so the external particles are identical. In this case, CP conser- vation requires an extra factor of i in the coupling to weak interactions. The posi- tively charged current and its hermitian conjugate are not isospin partners in the same way as the first class currents, but rather differ in sign. This current cannot have a neutral partner, as such a neutral second class current would be antihermitian. The strength of the coupling is, a priori, a free parameter. There is no obvious universality principle to fix the strength because the second class current is not a generator in an SU(2) algebra as is the first class current, nor does it commute with the first class SU(2) as does the leptonic current. Therefore we write the total weak current as J :J~+ +J(12 + iKJ (2) h+ ' j4f = j~_ + j(~ _ iK J(2) , (1) (where j(2) will be chosen with some simple normalization). J~ is the leptonic cur- M.-X Chen et al. / Second class currents 149 rent and J(h 1) the first class hadronic current. The simplest choices of K seem to be K = 0 or K = +1. Our main result in this paper is to determine the value of K in terms of the model for j(2) and the experimental data. In the next section we present the model and calculate K. In sect. 3 we turn to applications of our model to particle experiments, and present our conclusions in sect. 4. 2. Model for second class currents The second class currents, V and A, are constructed in this model from the first class currents and the pseudoscalar mesons, n, as follows = v + A , (2) Va**- 1 2<0> dec(Ga (3) AAUa _ 1 X/~ ClabcOrb V, + VU~b) (4) 2(o) c ' where the Latin indices run from 0 to 8, and (o) is the vacuum expectation value of the singlet o-field in the SU(3) X SU(3) o-model. The axial current, but not the vec- tor current, is of phenomenological interest *. In the o-model the commutator of the axial charge and ft involves products of o-fields and V. Following Adler et al. we replace the o-fields by their vacuum expectation value: % -+ ~Sa0(O} . (5) I11 doing so we obtain the connnutation relation 1 \[Q~, A~\] =idabcVc~ +-f~V/~dbcd face (rr d AU+A~e~d)e . (6) From the first tern\] of this equation, the simplicity of the choice of normalization in eqs. (2)-(4) is evident; this defines the strength K of eq. (1) in an appropriate way. Eqs. (3), (4) and (6) involve products of operators. The matrix elements of such products between vacuum and single-particle states are convergent only if cut-off form factors are introduced and they are linearly dependent upon the cut-off param- eters. Since the relevant form factors are not well known these equations are not very convenient for the evaluation of matrix elements of the second class current. Since we will assume B-dominance of the second class current, we are interested in the coup- * In coupling nucleons to leptons, the 2nd class vector current is proportional to the lepton mass. There is no 2nd class vector meson which can be diffractively produced. 150 M.-S. Chen et al. I Second class currents ling between \]1 -~ and B, which is defined as %1~10>: if B~, (7) where e~ is the polarization vector of the B-meson. To avoid the rather sensitive de- pendence on the cut-off, we decompose the second class currents according to their strangeness contents by defining where 1 ~ di/k( 5. V~. + V.U~rk), ~;.u = 2 (o---~ Then (8) (i,j,k:O, 1,2,3,8). ~"u =_ 1 i 2(o>x/~2dia~(zro~V~+V~r3 )' (??'13 : 4' 5' 6' 7) ' ,~u obeys a simpler commutation relation (!) \[Q5,27 1 = iVU w , (9) (1o) (11) where c~+_ _~ x/}-(c'~ 1 + c~2 ) for any operator c~ and VUw =x/~g V~ -V~-~ V~. Since eq. (11) no longer involves products of currents, we can derive a usual current alge- bra type sum rule to obtain fl~, which is defined as '" . (12) <~ i.7~" i0): ifBe B gab Then fB can be calculated from f13 by taking a ratio of an approximate evaluation of the right-hand sides of eqs. (9) and (10). Using standard current algebra techniques \[7\], we consider TUV(p, q2) =_ f eiqX O(xo ) (w(P) L \[AU_(x), A~(0)\] 10) d4x, (13) UV(u, q2) =_feiqx O(xo ) (o:(/91 \[O AU_(x), A~(0)\] 10) d4x, (14) where v = p" q and U v can be written as Uv(v, q2)= U(1)(v, q2) e~ + U(2)(u, q2)e v . q pV + U(3)(u, q2)e ~ . q qV. (15) T uv and U v are related to each other by + u ~ +fd4x eiq x iq T uv 8(x o) x <~(p) i \[A ?_ (x), ~7(0)I I0 >: o. (i 6) From eq. (8) and the fact that T uv has no pole atq u = 0, we obtain U(1)(0,0) = ifw , (17) M.-S. Chen et al. / Second class currents 151 where fw is defined by (col V~ 10) : if w e~ (which differs by a factor of 3 from the conventional choice). The right-hand side of eq. (14) can be evaluated by a disper- sion relation giving U(1)(0,0) = - ~f V(1)(~ -''0) du' , (18) /2 where VV(u',O)=(27r) 4 ~ (col/) A u In)(n\] ~,vlO)~4(p+q, pn), (19) n # - (27r)3 2E z and V (1) is defined in terms of V v similarly to U (1) in eq. (15). The lowest single- particle state for n is the B + meson. Thus B+saturation of eqs. (18) and (19), and PCAC, gives s;4%~. ~(m~_m~) =&, (20) where we have used PCAC in the form (col ~uA ~'_ IB+) =@-~ GB~eco " e B ? (21) Withf n = 0.95m~, m B -- 1.23 GeV, m~ = 0.78 GeV, and GBw~r -- -4.1 GeV \[8\] in eq. (20), we have f~ = -2.35f? ? . (22) We write the full coupling strength, fB; as fB = (fB/J'i3) f13- In the SU(3) symmetry limit the ratio fB/?g is determined entirely by the values of dab c in eqs. (9) and (10), plus the fact that the BTr V coupling is SU(3)symmetric: B + alTrbVU\]O)cCdabc (23) This ratio is fB = 1.5f~, (exact SU(3)). (24) In the following, in order to allow for SU(3) breaking, we perform an explicit calcu- lation of the proportionality constant between fB and 3'~ from eqs. (9)and (I0), with single-particle saturation and a cutoff. Eqs. (9) and (12) can be rewritten as # , 1 %~ = <~+l 2-~o > vq (~+v~ + v~ + + ~v~ + v~)10>. (25) Saturating eq. (25) with single-particle states gives u '~ 1 ~ d3pco %f; 2(o) ~-,If)2~-L" (B+S~+lco>(col v~10> Tf 152 d3pTr +f (2rr) 3 2E~ M.-S. Chen et al. / Second class currents (B+ l V~ l~ ) p / (rnB-m co) 2 1 - dq2 / J~GBwTr : 16rr 2 m2 "t m2 - q2 _ ie X \[(m 2 + m2co _q2)2 _4m2m2\]l/2 +(co++rr)} +(rr-+r?,co~ p). (27) Eq. (27) would be linearly divergent if no q2 cutoff (form factor) is introduced for GBcorr and/or fo0; therefore we simply replace fwGBwrr by its on-mass-shell value multiplied by a dipole form factor (1 -q2/A2)-2. We should also point out that, in this explicit calculation, fB acquires an imaginary part from the fact that the B + mass is above the corr threshold. Both the magnitude and phase offB approximately have linear dependence on A 2 . Using (o) = --mNgA/g r ~ --0.087 GeV we obtain f~ --~ -(2.5 + 0.8i)fw for A = 1.75 GeV. The full coupling strengthf B can be calculated in a similar manner by including the K*+K 0 contributions in eq. (10). Using the same value of A, we obtain fB = -(3.8 + 0.80 fco and therefore IfBI ~ 1.41f~ \[. This pro- portionality constant has very little A-dependence and approaches the SU(3) sym- metric value of 1.5 for large values of A. Thus we conclude that & --3.3f0o. (2s) At this stage, let us comment on other models for obtaining fB" In the present model, the commutator of two first class currents and that of two second class cur- rents are not equal to each other and there are no Weinberg sum rules to relate fB and fo" There exist other models \[9-11 \] in which fB = fo by generalized Weinberg sum rules. However, in the latter models the right-hand side of eq. (6) only has a term ifab c Vc u instead of the first class currents. For such models our PCAC sum rule would lead to fl~ = 0. The inconsistency of these models with PCAC is one reason why we have concentrated on the model given in eqs. (3) and (4), even though it may not be impossible to introduce new kinds of PCAC anomalies or other modifi- cations to the other models to resolve the inconsistency. Now we return to deduce the consequences of our approach. Having obtained fB, we can calculate the coupling strength of the second class current to the nucleons by vector dominance, which is a good approximation at low momentum transfer for the first class currents. We parametrize the second class coupling of a nucleon to the weak interactions as < p(e2 ) l KJ~h2) l n(P 1) > = < p(Pz ) l K 3" l n(Pl ) > NP ? = -i~(e 2) o -- 75 gH u(P1) (29) uv 2m M.-S. Chen et al. / Second class currents 153 where q = PI - P2" B+ dominance ofA ~ gives ! gn 1 K/13 gBNN (30) 2m- gBN q2 where gBNN is the B-meson nucleon coupling constant and is determined from strong interaction Regge exchanges * as gBNN ~ 22.8 GeV -1 . With fB ~- --3"3f~o, fa) = 0.1 GeV 2, we obtain I glI ~ -9.4K ~-7.7Kg A . (31) From the results of the most recent experiment \[1\], Holstein and Treiman \[4\] ex- t tract the value gll = (-8 -+ 3) gA" Thus, from eq. (31) we find as the principle result of this paper, that K = 1.03 -+ 0.40, (32) (K was defined in eq. (1)). In addition to the statistical error quoted, there are un- P known errors due to the dependence on the model used to extract gll from the nu- clear physics data, and due to the extrapolation ofgBN N from the Regge trajectory toy = l. The consistency of our value for K with unity, given the experimental result of ref. \[1\], suggests some sort of universality, although as discussed in sect. 1 there is no obvious universality principle to impose. Moreover, earlier experiments \[14\] would give the values K ~ 0.5 or K ~ 0. Our model can be used for computing second class effects in any hadronic pro- cess. Our result, assuming K = 1, is (fill \[i) =-0.23(fIB \]i), (33) where (fiB s l i) is the coupling of the B-meson to the states whose current is de- sired. This coupling is to be deduced from strong interactions, perhaps from B-meson Regge exchanges. The total weak current is jweak = j~ + g + A + A + P . (34) # ~t /.t # # /.t V u in its usual couplings is proportional to qu' which in a leptonic process is pro- portional to the lepton mass, and can therefore be neglected. For that reason we have ignored it. In the next section we apply our model to two interesting cases in particle phys- ics, B-meson production by neutrinos and A production by neutrinos. Holstein and Treiman have emphasized the importance of this latter process. The coupling gB+N~ is probably rather well known as its ratio to the 7rNN coupling is tightly constrained in ref. \[12\] by the ZGS data on polarized proton total cross-section differences \[13\], aT(t ~) - at(it). The precise relation of the present coupling to that of ref. \[12\] is ,2 1/2 gB+N~ = w/2gBo Nfq = x/~(GBON~/2So) = GBON~I/(1 GeV). 154 M.-S. Chen et aL / Second class currents 3. Appncations 3.1. B-oroduction If there are second class currents, the reaction vp -+/a-B+p, or the same reaction on nuclei, will occur diffractively. According to our model, the rate of this process should be rather large compared to the diffractive production of other vector mesons. With K = 1, the matrix element gives 'M(UPlM(pp__>p_p+p)/-+ #- B+p)I2 ~ (?T(BN)fB/m2B )2 \OT(PN) tL,/rn---~ ,~ 1.7(OT(BN)/OT(PN))2 , (35) aside from expected differences in the Q2 dependence of the B and O amplitudes. At lower energies this ratio is reduced significantly by phase space, but we expect the B- production to be comparable to p production assuming OT(BN ) ~ oT(PN ). Unfortunately, we cannot just nmltiply existing estimates of p production in u reactions by the appropriate factor, because all such estimates we are aware of make use of models which are in strong disagreement with data for electroproduction of p's. We are preparing a paper in which we calculate the diffractive production of all vector mesons, in both electron (ninon) reactions and u reactions, checking that we agree with experiment in the former. In that paper we will present detailed predic- tions for the diffractive B-production. If experimental data on diffractive production of p's in u reactions becomes available, we predict a similar cross section will be found for B-production given our model and the data of ref. \[ 1 \]. 3.2. A production Holstein and Treiman \[4\] suggest a comparison of the processes vp ~/~-~+p, tPn -+ ~t+Tr- n, to find second class effects. Such effects show up most strongly in a difference be- tween the coefficient of a particular angular dependence in the two processes \[15,16\] *. They further suggest looking at the A mass region in order to enhance this effect. With a particular model for this process they find a large effect. We find, however, that with a B-dominated second class current, this effect de- pends on background and/or small terms, and therefore that its magnitude cannot be easily used to measure the strength of the second class current coupling. Our argument is as follows. The cross section of interest is, if the A is dominant, an interference between two amplitudes in which the A's have a difference of 2 units of spin projection in their rest frame with the z-axis along the nucleon \[16\]. The dominant coupling of a B, and therefore of a B-dominated second class current, to It is a 03-1 in the notation of ref. \[16\]. M.-X Chen et aL /Second class currents 155 an N ~ 2x transition is expected to be the amplitude exchange degenerate to the pion. This amplitude is -- (36) where N, A v are the nucleon and 2x spinors, P = P~ + PN, and Q = PA -- PN" Then eu(B)M u is just proportional to the pion emission amplitude &~Quu. The spinor is made of a spin-? spinor and a polarization vector: : <{ M m I M> . m In this frame, Q has only longitudinal components, so Q. e =P 0 only for m = 0. In this frame ONhas no spin dependence so u has the same spin as the nucleon. There- fore, the second class current is spin non-flip, assuming eq. (36) gives the form of the dominant current. The dominant first class currents are of the form ~u(gA +gv T5 )N" The vector current then is mainly a magnetic dipole. The axial current by PCAC is really mainly &u(6vv - QvQv/(Q 2 + M2))N so that it is conserved as M2~ -+ 0, and the term with Q~ gives a cross-section contribution proportional to the lepton mass so it can be ignored. The quantity ON is spin independent and 61'5 N oc o z in the frame in which we are working, so O has the same spin as N. Therefore the dominant first class current can contribute at most one unit of spin flip, from e v alone. Since the dominant second class current does not flip spin, the only spin flip two terms come from the first class current twice. Thus no second class effects from the dominant terms contribute to the angular dependence of interest. Consequently, an observation of such effects in v(vON ~ txA would verify the exis- tence of the second class current, but would not provide a model-independent deter- ruination of its strength. Only the diffractive B-production could do that. Our result differs from that of ref. \[41 because of the model used for A produc- tion. We proceed phenomenologically and assume that BNA current has the form sug- gested by exchange degeneracy with the 1rNA current; this chooses one of the four possible terms for the current. If the second class current had a dynamical origin re- lated to the properties of the B, this would be sensible. Alternatively, Holstein and Treiman use the second class nucleon current and directly obtain the A production in terms of that from Adler's model for A production. This populates invariants dif- ferently from our approach and assumes that different dynamical effects are impor- tant. This gives a significant cos 2~0 angular dependence. Thus experimental observa- tions in A production by neutrinos will clarify the situation about the dynamics of second class currents. The observation of a large cos 2tp effect will give evidence for the existence of second class currents and will favor the Holstein and Treiman approach. The absence of a cos 2~ effect plus independent confirmation of the existence of sec- ond class currents will favor the B-dominance approach. 156 M..S. Chen et al. / Second class currents 4. Conclusions If the present experimental results are confirmed and it remains true that the strength of the second class coupling to weak interactions is the same as that of first class currents (which we describe as K = 1), it will be difficult to ignore second class currents in constructing a fundamental weak interaction theory, even though no at- tractive way to do so has appeared. Approaches to symmetries may appear (e.g. ref. \[17\] ) in which the B vector meson plays a more basic role and in which universality takes on a more general meaning. Perhaps there is a relation between the strong inter- action fact that the B-meson exists, and the possible weak interaction fact that a cur- rent with the B's quantum numbers plays a significant role. The crucial particle physics experiments to verify second class currents are A pro- duction and B-production by neutrinos. Unfortunately, our model suggests that there is no simple relation between the strength of the second class coupling and the size of the effect in A production. On the other hand an observation of B-production in up -+/a-B+p or related reactions would provide a rather direct measurelnent of the second class coupling, which we predict to be Kf B = 3.3f o. We are grateful to J. Luthe, for numerically discovering the absence of second class effects in the dominant A amplitudes, to Y.-P. Yao for helping clarify the theoretical properties of second class currents, and to B. Roe for discussions concerning the pro- duction of B mesons in u reactions. References \[1\] F.P. Calaprice, S.J. Freedman, W.C. Mead and H.C. Vantine, Phys. Rev. Letters 35 (1975) 1566. \[2\] S. Weinberg, Phys. Rev. 112 (1958) 1375. \[3\] S. Adler et al., Phys. Rev. D12 (1975)3522. \[4\] B.R. Holstein and S.B. Treiman, Princeton Univ. preprint (1975). \[5 \] L. 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